Question: Mustafa is staring at a pizza spinning on a wheel in a glass case. The pizza is cut into four even slices. The distance $P(t)$ (in $\text{cm}$ ) between the center of the tastiest looking slice and the door of the glass case as a function of time $t$ (in seconds) can be modeled by a sinusoidal function of the form $a\cdot\cos(b\cdot t)+d$. At $t=0$, the center of the tastiest looking slice is farthest from the door, at a distance of $30\text{ cm}$ away. After $2\pi$ seconds, it is closest to the door, at a distance of $10\text{ cm}$. Find $P(t)$. $\textit{t}$ should be in radians. $P(t) = $
Solution: The strategy First, we should convert the given information about the real-world context into mathematical terms of the sinusoidal function and its graph. Then, we should use the given information to find the amplitude, midline, and period of the function's graph. Finally, we should find $a$, $b$, and $d$ in the expression $a\cos(b\cdot t)+d$ by considering the features we found. Converting the given information into mathematical terms At $t=0$, the center of the slice is $30\text{ cm}$ from the door. This means the graph of the function passes through $(0,30)$. We are given that this is the farthest point from the door, which corresponds to a maximum point of the graph. $2\pi$ second later (which means $t=2\pi$ ) the distance is $10\text{ cm}$. This corresponds to the point $(2\pi,10)$. We are given that this is the closest point to the door, which corresponds to a minimum point of the graph. In conclusion, the graph has a maximum at $(0,30)$ and then has a minimum point at $(2\pi,10)$. Determining the amplitude, midline, and period The midline intersection is halfway between the maximum and minimum, which is at $y={20}$, so this is the midline. The minimum point is $10$ units below the midline, so the amplitude is ${10}$. The minimum point is $2\pi$ units to the right of the nearest maximum, so the period is $2\cdot 2\pi={4\pi}$. [Why did we multiply by 2?] Determining the parameters in $a\cos(b\cdot t)+d$ Since the maximum at $t=0$ is followed by a minimum point, we know that $a>0$. [How do we know that?] The amplitude is ${10}$, so $|a|={10}$. Since $a>0$, we can conclude that $a=10$. The midline is $y={20}$, so $d=20$. The period is ${4\pi}$, so $b=\dfrac{2\pi}{{4\pi}}=\dfrac{1}{2}$. The answer $P(t)=10\cos\left(\dfrac{1}{2}t\right)+20$